×

Strong asymptotics of two-point Padé approximants for power-like multivalued functions. (English. Russian original) Zbl 1300.41009

Dokl. Math. 89, No. 2, 165-168 (2014); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 455, No. 1, 138-141 (2014).
The authors study two point Padé approximants constructed from two germs \( \omega _{0}\) and \(\omega _{\infty }\) of the same multivalued analytic function of the form \[ w\left( z\right) =w\left( z;\mathcal{A},\alpha \right) :=\prod \limits_{j=1}^{\infty }\left( z-a_{j}\right) ^{\alpha _{j}}, \] where \(p\geq 2,\) \(\alpha _{j}\in \mathbb{C}/\mathbb{Z}\), \(\sum_{j=1}^{p}\alpha _{j}=0\), \(\alpha _{1},\alpha _{2},\ldots,\alpha _{p}\) are any different points in \(\mathbb{C}^{\ast }=\mathbb{C}/\left \{ 0\right \}\), \( \mathcal{A=} \{ a_{1,}a_{2},\ldots,a_{p}\}\), and \(\alpha =\{ \alpha _{1},\alpha _{2},\ldots,\alpha _{p}\}\). They obtain a second order differential equation with polynomial accessory parameters and solve the equation.

MSC:

41A21 Padé approximation
41A63 Multidimensional problems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Aptekarev, A I; Buslaev, V I; Martinez-Finkelshtein, A; Suetin, S P, No article title, Russ. Math. Surveys, 66, 1049-1131, (2011) · Zbl 1242.41014
[2] A. I. Aptekarev and M. L. Yattselev, arXiv:1109.0332.
[3] Buslaev, V I; Martinez-Finkelshtein, A; Suetin, S P, No article title, Proc. Steklov Inst. Math., 279, 25-51, (2012) · Zbl 1298.30028
[4] Buslaev, V I, No article title, Sb.: Math., 204, 190-222, (2013) · Zbl 1276.41011
[5] Fedoryuk, M V, No article title, Itogi Nauki Tekh., Ser.: Sovr. Probl. Mat. Fundam. Napravleniya, 13, 93-210, (1986) · Zbl 0655.41034
[6] Laguerre, E, No article title, J. Math. Pures Appl., 1, 135-165, (1885)
[7] Martinez-Finkelshtein, A; Rakhmanov, E, No article title, Commun. Math. Phys., 302, 53-111, (2011) · Zbl 1226.30005
[8] Martinez-Finkelshtein, A; Rakhmanov, E A; Suetin, S P, No article title, Contemp. Math., 578, 165-193, (2012) · Zbl 1318.42033
[9] Martinez-Finkelshtein, A; Rakhmanov, E A; Suetin, S P, No article title, Russ. Math. Surveys, 68, 183-185, (2013) · Zbl 1275.41019
[10] Nuttall, J, No article title, Constr. Approx., 2, 59-77, (1986) · Zbl 0585.41014
[11] Stahl, H, No article title, J. Approx. Theory, 91, 139-204, (1997) · Zbl 0896.41009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.